Poisson Distribution Calculator

What Is Poisson Distribution?

This distribution helps to predict the probability of how many times a specific number of events can occur within a fixed interval (space or time). 

Example: Imagine counting the number of people passing through a walkthrough gate in one minute. Poisson distribution helps determine the probability of a specific number of people passing through during the defined duration.

Properties of Poisson Distribution:

• All events occur independently of each other.
• Two events cannot occur at the same time.
• Mean: \(E(X) = \lambda\) and Variance: \(V(X) = \lambda\).
• The average rate of occurrence (\(\lambda\)) remains constant over time, where \(n p = \lambda\).
• The standard deviation is the square root of the mean: \(\sigma = \sqrt{\lambda}\).

Poisson Distribution Formula:

\[ P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} \]

Where:

• \(P(X = x)\) is the probability of \(x\) occurrences
• \(e\) is Euler's constant (\(\approx 2.71828\))
• \(\lambda\) is the average rate of occurrences
• \(x\) is the number of occurrences (Poisson random variable)
• \(x!\) is the factorial of \(x\)

How To Calculate Poisson Distribution?

• Determine the average rate of occurrences
• Write down the desired number of occurrences (x)
• Calculate the factorial of x 
• Put values in the Poisson distribution formula, solve the exponent part
• After that divide the result by the factorial of x

Example:

Suppose you work in a call center where you receive an average of \(\lambda = 4\) calls per minute. Calculate the following probabilities:

• \(P(X = 3)\): Probability of receiving exactly 3 calls in a minute
• \(P(X < 3)\): Probability of receiving less than 3 calls in a minute
• \(P(X \le 3)\): Probability of receiving at most 3 calls in a minute

Solution:

1. Probability \(P(X = 3)\)

Using the Poisson formula:

\[ P(X = 3) = \frac{e^{-\lambda} \lambda^3}{3!} = \frac{e^{-4} \cdot 4^3}{3!} = \frac{0.018315 \cdot 64}{6} \approx 0.19536 \]

So the probability of receiving exactly 3 calls per minute is approximately \(19.54\%\).

2. Probability \(P(X < 3)\)

We sum the probabilities for \(X = 0, 1, 2\):

\[ \begin{aligned} P(X = 0) &= \frac{e^{-4} 4^0}{0!} \approx 0.01832 \\ P(X = 1) &= \frac{e^{-4} 4^1}{1!} \approx 0.07326 \\ P(X = 2) &= \frac{e^{-4} 4^2}{2!} \approx 0.14652 \end{aligned} \]

Adding them:

\[ P(X < 3)=P(0) + P(1) + P(2) \approx 0.01832 + 0.07326 + 0.14652=0.2381 \]

Hence, the probability of receiving less than 3 calls per minute is approximately \(23.81\%\).

3. Probability \(P(X \le 3)\)

We sum the probabilities for \(X = 0, 1, 2, 3\):

\[ P(X \le 3) = P(0) + P(1) + P(2) + P(3) \approx 0.01832 + 0.07326 + 0.14652 + 0.19536 = 0.43346 \]

Thus, the probability of receiving at most 3 calls per minute is approximately \(43.35\%\).

Calculating Poisson probabilities manually can be time-consuming. To save time and simplify the calculation use our poisson distribution calculator. No matter, whether you are a beginner, student, researcher, or professional, the calculator can handle all your Poisson probability needs.

Poisson Distribution Table:

What Is The Difference Between Poisson Distribution And Binomial Distribution?

Poisson Distribution:

• The variance is equal to the mean
• The number of event occurrences is counted over a fixed time interval or space 
• This is suitable for events happening independently at a constant rate

Binomial Distribution:

• Each trial has Two possible outcomes (success or failure)
• The number of times an experiment is repeated is known

When Do We Use Poisson Distribution?

Poisson distribution is good for modeling independent events at a constant average rate within a specified interval. 

Here are some general use cases:

• Counting occurrences 
• Rare events 
• Quality control
• Queueing systems

Easily calculate Poisson probabilities for these scenarios with the help of our Poisson distribution calculator. It can handle a variety of use cases, providing reliable results.

Reference:

From the source of Wikipedia: Probability mass function, Assumptions and validity. 

From the source of Investopedia: Understanding Poisson Distributions.

From the source of Brilliant ORG: Conditions for Poisson Distribution, Probabilities, Properties.